On the representation of a class of contractive matrix-valued functions (Q1083670)

We consider in this article the class S of contractive matrix-valued functions S(z) of order n, that are holomorphic in the unit disk, for which \(\| S(z)\| \leq 1\). From this class we study the subclass \(S\Pi\), defined by Arov, of functions whose boundary values a.e. are, simultaneously, boundary values of matrix-valued functions \(\tilde S(z)\) meromorphic outside the unit disk, with elements of bounded characteristic there i.e. \(\lim_{| z| \uparrow 1}S(z)=\lim_{| z| \downarrow 1}\tilde S(z)\) a.e. A decomposition of \(L_+^ 2(C^ n)\) in two orthogonal subspaces, of order k and n-k, is established by constructing two sets of functions \(\{\Phi_ m(\xi)\}^ n_{m=1}\) and \(\{\Psi_ m(\xi)\}^ n_{m=1}\) \((\xi =e^{it}\), \(0\leq t\leq t2\pi)\) which are, for each fixed \(\xi\) except possibly a set of zero measure, eigenvectors of the nonnegative matrices \(I_ n-S^*(\xi)S(\xi)\) and \(I_ n-S(\xi)S^*(\xi)\), respectively (* denotes Hermitian conjugation), and orthonormal basis of \(C^ n\). The above mentioned decomposition allows us to obtain the following representation for S(z)\(\in S\Pi\) \[ S(\xi)=U_ 2^{- 1}(\xi)\left( \begin{matrix} S_ 1(\xi)\\ 0\end{matrix} \begin{matrix} 0\\ S_ 2(\xi)\end{matrix} \right)U_ 1(\xi)\quad a.e., \] where \(U_ 1(\xi)\) and \(U_ 2(\xi)\) are a.e. limiting values of inner matrix-valued functions, and the block \(S_ 1(\xi)\) and \(S_ 2(\xi)\), of order k and n-k respectively, verify a.e. the relations \(\| S_ 1(\xi)\| <1\); \(\| S_ 2(\xi)\| =1\). On the basis of these representations it is possible to study the problem of multiplicity of a scattering matrix S(z) of a linear passive n-port when S(z) is not inner and satisfies the condition det [I\({}_ n-S^*(\frac{1}{\bar z})S(z)]=0\) in the unit disk.